Ferkans — Interactive Telecom Tutor

From Eigenbeams to User Beams

The SVD of $\mathbf{H}_1$ gives us $r$ eigenbeams — natural modes of the BS-RIS channel. But users live in specific directions, not in eigenbeam directions. Section 11.3 shows how the RIS phase-shift matrix $\boldsymbol{\Phi}$ maps eigenbeams to user beams. The RIS operates as a beam steering matrix: input is the set of eigenbeams (coming from the active array), output is the set of focused beams toward users.

Theorem: Eigenbeam-to-User-Beam Mapping via $\boldsymbol{\Phi}$

For the array-fed RIS, define per-user phase profile

$\boldsymbol{\phi}^{(k)} \in \mathbb{C}^N, \quad (\boldsymbol{\phi}^{(k)})_n = e^{j(\arg(\mathbf{h}_{k,2}^*)_n - \arg(\mathbf{u}_k)_n)}.$

This aligns the $k$ -th eigenmode's RIS-side signature with the direction of user $k$ .

The composite RIS phase is a weighted superposition

$\boldsymbol{\phi}^\star = \frac{\sum_k \alpha_k \boldsymbol{\phi}^{(k)}}{\|\sum_k \alpha_k \boldsymbol{\phi}^{(k)}\|}\ \text{(element-wise normalized)}$

where $\alpha_k$ are user-specific weights. Under orthogonal eigenmodes (from SVD), the user beams are nearly orthogonal, so cross-interference is small.

For $K \leq N_t = r$ users, each assigned one eigenmode: per-user SINR is $\approx \sigma_k^2 N / K$ with negligible inter-user interference.

Pick one eigenmode per user. The active array sends signal through the $k$ -th eigenmode, producing a field at the RIS that is proportional to $\mathbf{u}_k$ . The RIS then focuses that field toward user $k$ 's direction via a phase profile $\boldsymbol{\phi}^{(k)}$ . Multi-user case: superpose the phase profiles with suitable weights.

Proof

Per-eigenmode beam

Signal in eigenmode $k$ : $\mathbf{u}_k \cdot x_k$ at the RIS. Reflected: $\boldsymbol{\phi}^{(k)} \odot (\mathbf{u}_k x_k)$ . At user $k$ : $\mathbf{h}_{k,2}^H (\boldsymbol{\phi}^{(k)} \odot \mathbf{u}_k) x_k = \mathbf{h}_{k,2}^H \text{diag}(\boldsymbol{\phi}^{(k)}) \mathbf{u}_k \cdot x_k$ . With $\boldsymbol{\phi}^{(k)}$ matched, magnitude is $\|\mathbf{h}_{k,2}\|_1 \cdot \|\mathbf{u}_k\|_1 \cdot 1/\sqrt{N} \sim \sqrt{N}$ per element, $N$ coherent sum.

Eigenmode orthogonality

Different eigenmodes $\mathbf{u}_k, \mathbf{u}_j$ are orthogonal: $\mathbf{u}_k^H \mathbf{u}_j = \delta_{kj}$ . Hence eigenmode $j$ at user $k$ has gain $\mathbf{h}_{k,2}^H \text{diag}(\boldsymbol{\phi}^{(j)}) \mathbf{u}_j = 0$ in the ideal case (minus corrections).

Sum rate

Under ideal eigenmode and RIS phase design, each user sees its own eigenmode with full $N$ coherent gain. Sum rate scales linearly in $K$ (multiplexing) and logarithmically in $N$ (aperture). $\blacksquare$

Can We Serve More Users Than Eigenbeams?

Eigenbeam count = $r = \min(N_t, N_{\text{eff}})$ . What if $K > r$ ? Three options:

Time-sharing: serve at most $r$ users per slot, rotate across time. Loses a factor of $K/r$ in per-user rate.
Non-orthogonal multiplexing (NOMA): superpose users on the same eigenmode with decoding order. Loses some rate to successive interference cancellation imperfections.
Upgrade the array: increase $N_t$ to match $K$ . Expensive at mmWave; but increasing RIS size doesn't help (bottlenecked by $N_t$ ).

For modest user counts ( $K \leq 8$ ), option 3 is typical: match $N_t$ to $K$ . This is why the CommIT array-fed RIS typically targets $N_t \sim 8$ - $16$ and $K \sim 4$ - $8$ — consistent sizing.

Array-Fed RIS Sum Rate vs. $K$

Plot sum rate as a function of $K$ for array-fed RIS with different $N_t$ . The sum rate grows linearly in $K$ until $K = N_t$ , then saturates (or degrades with time-sharing / NOMA). The "knee" at $K = N_t$ is the multiplexing limit.

Parameters

Active antennas

N_t

8

RIS elements

N

256

Max users

K

12

Transmit SNR (dB)10

Example: User-Eigenmode Assignment

$N_t = 4$ active antennas, $N = 128$ RIS elements, $K = 4$ users at diverse angles. Describe the eigenmode-user assignment process.

Solution

SVD

Compute SVD of $\mathbf{H}_1 = \mathbf{U}_1 \boldsymbol{\Sigma}_1 \mathbf{V}_1^H$ ; extract $r = 4$ eigenmodes.

Per-user eigenmode scores

For each (user $k$ , eigenmode $j$ ) pair, compute $s_{k,j} = |\mathbf{u}_j^H \mathbf{h}_{k,2}^* / \sqrt{N}|^2$ — the expected per-mode SNR contribution.

Optimal assignment

Solve Hungarian (assignment) algorithm: match users to eigenmodes such that $\sum_k s_{k, \pi(k)}$ is maximized. Polynomial-time, optimal.

Result

Each user $k$ is assigned one eigenmode $\pi(k)$ , typically the one most aligned with user $k$ 's direction. Active precoder: $\mathbf{v}_{k} = \mathbf{v}_{\pi(k)}$ . RIS phases: compute per-user $\boldsymbol{\phi}^{(k)}$ and combine. $\blacksquare$

Array-Fed RIS: Eigenmode → User Beam Mapping

Animation showing the SVD decomposition of the near-field BS-RIS channel, the resulting eigenbeam "signatures" on the RIS surface, and how the phase-shift matrix

\boldsymbol{\Phi}

steers each eigenbeam toward a different user.

Why This Is Exactly Right for mmWave

At sub-6 GHz: small active arrays are cheap, and conventional massive MIMO already provides many DoF — array-fed RIS is a marginal improvement.

At mmWave (28–100 GHz): active arrays are expensive per antenna (each RF chain has a high-power GHz-class ADC+DAC), but physical aperture is cheap (metasurfaces are inexpensive per unit area). The array-fed RIS shifts the cost from expensive RF chains to cheap metasurface elements.

At sub-THz (100–300 GHz): even more extreme. Active array cost per element scales poorly; RIS cost per element scales well. The architecture is arguably the only practical way to deploy multi-user multiplexing at these frequencies.

This is the central design philosophy of the CommIT array-fed RIS: use cheap passive aperture to amplify the capability of expensive active hardware.

Multi-User Multibeam Operation